Dembski goes mathematical

By Mark Perakh

Critics of William Dembski's publications often pointed to the absence of real mathematics in his multiple books and papers while his colleagues often acclaim his supposed mathematical rigor to the extent of comparing him to great scientists of the past (as, in the much derided example, when philosopher Robert Koons bestowed on Dembski the title of Isaac Newton of information theory). Dembski has been very selective in choosing when to respond to his critics and when to ignore the critique. Usually he promptly responded to those critics who do not sport high-level degrees but are either students or possess a Bachelor's, or at the most a Master's degree, and pointing to this irrelevant factoid as though it indeed determined the quality of their (usually well substantiated) critical comments. He has studiously avoided responding to those critics who, like for example famous mathematician David Wolpert, could not be denigrated for alleged lack of qualifications.

However, while keeping the facade of invulnerability to critique, apparently Dembski felt uncomfortable with the obvious lack of respect of his work by professional mathematicians and other real scientists. So recently Dembski emailed a group of a dozen or so of his critics an article wherein he suggested an allegedly novel measure of information which he labeled Variational Information. This article can be seen at http://www.iscid.org/ubb/ultimatebb.php?ubb=get_topic;f=6;t=000533. As is usual for Dembski, this article is full of esoteric notations incomprehensible to average readers but seems to reveal his reasonable familiarity with the mathematical parlance and generally with various subfields of mathematics, including information theory and related fields. Until now, however, most of what Dembski proposed as a rigorous mathematics looked more like what I labeled "mathematism" (see www.talkreason.org/articles/dembski.cfm). This term denotes the use of mathematical symbolism for the sake of symbolism without real mathematical rigor and without any evident purpose except, perhaps, for trying to impress a wide audience with the supposed sophistication of what, in fact, more often than not lacked any substance.

Is this new article different?

It did not take long for several of Dembski's critics noticed weaknesses in Dembski's new article, in particular the lack of innovations despite Dembski's claim to the contrary. A brief discussion took place on some internet fora, where various faults of Dembski's paper were pointed to. In the course of these discussions we came to a consensus that a critical review of Dembski's new paper shall be entrusted to a professional mathematician from among the fora's particiapnts, although representatives of other professional fields would be free to suggest ideas to be included in the planned critical review.

However, before the plan of preparing the review in question came to fruition it has been made largely unnecessary. A highly qualified expert in information and complexity, Cosma Rahilla Shalizi, posted to the web a devastating critique of Dembski's new mathematical exercise, preempting most of the critique planned for the critical review in question. Additionally, a partial critique of certain elements of Dembski's math was posted by another well-qualified critic, David Wilson, who pointed to certain errors in Dembski's derivations.

I believe the judgment on Dembski's new allegedly rigorous mathematical exercise is in. As Shalizi put it (using Dembski's own expression) the Isaac Newton of information theory has reinvented the wheel and demonstrated a level of ignorance of the relevant fields of information theory unforgivable for even a beginner trying his hand in this magnificent field of inquiry. In particular, as Shalizi has pointed out, the quantity which Dembski labeled Variational Complexity and claimed to be a novel measure of information estimating divergence between different probability
distributions is in fact a quantity known in information theory for over forty years - it is nothing but a well-known particular case of Rènyi divergence, namely Rènyi divergence of the second order. This quantity,
introduced in the early sixties by a survivor of the Holocaust, the brilliant Hungarian mathematician Alfrèd Rènyi (1921-1970), has been discussed many times in many papers published in international mathematical
journals and was applied to a variety of problems, both in information theory and beyond. Dembski does not ever mention Rènyi's name and provides no references either to the original papers and books by Rènyi or to any subsequent papers and books by many authors where Rènyi's ideas were discussed with many details. I don't know whether Dembski's claim to the discovery of a novel measure of information stemmed from his striking ignorance of the field in which he pretends to be an expert or if he knows the works of Rènyi and his followers but preferred to keep this knowledge to himself. In either case, there is no way Dembski can eschew a major embarrassment as the lack of any new math in his article comes now to light. Of course, for those familiar with Dembski's previous publications the above story brought no surprise - we would rather be surprised if Dembski had produced a real high quality mathematical paper - not because he is, in my view, incapable of doing good math, but because, as I see it, his overriding fervor to prove his beliefs blinds him to facts and logic.

Dembski announced that the paper in question is the first installment in a set of seven papers under the title "Mathematical Foundation of Intelligent Design." The first installment turned out to be a flop. As the list of the seven titles shows, among the planned articles there is one about the "law of conservation of information," another about the use of the NFL theorems for the support of intelligent design, and so on. Until now Dembski's attempts to introduce the law of conservation of information and to use the NFL theorems all turned out to be far from successful. As Dembski's new "mathematical" paper shows, he seems to be in a state of denial, largely ignoring the critique of his work, and continues in the same vein as a stubborn purveyor of faulty but pretentious claims. Judging from Dembski's new "mathematical" paper, there is good reason to expect that the "mathematical foundation of Intelligent Design" he plans will not only fail to support the edifice of a respectable theory but more realistically may be expected to collapse under its own weight.

Comment on September 12, 2004. This essay contains a link to Dembski's article. We have found that this link no longer opens Dembski's article in question (there is instead a message that the article will be re-posted when ist author is confident that he has "worked out all kinks"). The amended version of Dembski's article is, however, available at http://www.iscid.org/papers/Dembski_VariationalInformation_072404.pdf.